A musclebound Dane dangles from a pull-up bar on the second floor of the new Danish Architecture Centre, his bulging body straining in the window as visitors process up the stairs to see an exhibition on housing design. In another window nearby, a group of urban innovators conduct a meeting in a carpet-lined pod, while on the floor above, guests sit down to dine on smoked salmon smørrebrød prepared by the co-founder of Noma, overlooked by the roof terrace of swanky penthouse apartments.

Standing as a jumbled stack of museum, offices, gym, restaurant and housing, Copenhagen’s 2bn kr (£236m) Blox project is the latest attempt to achieve the ultimate architect’s dream: condensing the thrilling, messy energy of a city into a single building, creating a space for chance encounters between different people and activities. As a centre for architecture, it has a remit to demonstrate the power of the discipline to integrate these things, to prove that the whole can be more than the sum of its disparate parts. But does it pull it off?

Occupying the site of a former brewery, next to the windswept Kierkegaard Square, the sleek glass blocks join a range of other dark, glossy cultural buildings dotted along Copenhagen’s harbour front, from the neighbouring Black Diamond library to the royal theatre and opera house. While these palatial landmarks were designed by Danes, Blox is the work of the Office for Metropolitan Architecture (OMA), a practice founded by Dutchman Rem Koolhaas, who has been obsessed with the role of buildings as mixing chambers for multiple contradictory uses since he wrote his sizzling manifesto for urban intensity, Delirious New York, in 1978.

Photograph: Hans Werlemann/courtesy OMA

In this seminal book, he devoted a chapter to the wonder of the Downtown Athletic Club, a 1930s Manhattan skyscraper that combined swimming pools, boxing gyms, golf course, restaurants and apartments in a vertiginous layer cake. He rejoiced in the “fantastic juxtaposition of its activities” where each floor provided “infinitely unpredictable intrigue” and forced its users to “surrender to the definitive instability of life in the metropolis”. The building was a machine, he enthused, “to generate and intensify desirable forms of human intercourse”.

OMA has been trying to emulate the model ever since, with buildings that expose, provoke and create friction between their different users and functions, with mixed success. Its De Rotterdam project promised a dynamic integration of housing, offices and hotel, but ended up as three siloed towers on a plinth. The Timmerhuis, also in Rotterdam, was planned as a cloud of apartments suspended above a one-stop “civic agora” for council services, but the city pulled out and the great open forum was walled in. So has OMA finally succeeded in bringing a shock dose of multi-levelled metropolitan dynamism to the urbane context of Copenhagen?

From the outside, it doesn’t look promising. Far from suggesting unpredictable intrigue, the building exudes the off-putting sense of a generic corporate office block. Large expanses of green-tinted glazing stretch across the facade, punctuated by lighter bands of fritted glass and dark metal grilles, with no sign that there is anything remotely public going on inside – not even an entrance. Instead, visitors are are shuffled down stairs and escalators to an entrance foyer three storeys below ground. At street level, it is cars that take priority, driving along a main road that punches right through the middle of the building, beneath a blank undercroft. In this famously “liveable” pedestrian-friendly city, where half of the population commutes by bike, it is a most un-Danish prospect.

Photograph: Rasmus Hjortshøj/Coast

“It is an intentional critique of Danish urbanism,” says Ellen van Loon, OMA’s partner in charge of the project. “The city has become too calm and beautified. I miss the cars and the noise.” In keeping with their reputation as provocateurs, the Dutch architects have consciously made the least Danish venue imaginable for the Danish Architecture Centre. Where you might expect to find a warm world of blond wood, earthy baked brick and light-flooded white rooms, you are hit in the face with black concrete, black rubber flooring and raw steel, with exposed strip lights and industrial quantities of metal mesh – the recurring palette of many recent OMA projects.

“It feels like New York!” declares Kent Martinussen, director of DAC, who admits he tried to get the architects to use Denmark’s famous Petersen bricks or Dinesen floorboards early on, to no avail. “Some people think it is cold, but I prefer to say ‘cool’. It has a Lady Gaga vibe.” You can sort of see what he means, if Gaga had ditched the meat dress and set up a financial consultancy.

The corporate feel isn’t a surprise when you learn that the project’s client is Realdania, a private philanthropic association founded in 2000 on the sale of a bank and a mortgage institute, with assets of over 20bn kr (£2.4bn). Its primary interest in the building comes in the form of the “Bloxhub”, a high-end serviced office space that styles itself as a “matchmaking agency for built environment innovators”. At 6,000 kr (£700) per month, it is not a particularly cheap place to rent a desk (although discounts are offered to startups and flexible workers), but it has already attracted a host of local and global companies, no doubt enticed by the proximity to the national parliament.

“A launchpad for entering into the Nordic market,” is how Bloxhub director Torben Klitgaard describes it as we tour the jazzy open-plan world of pot plants, meeting booths and break-out think pods. Looking down on the DAC’s exhibition space, it becomes clear that the museum essentially occupies the atrium of a deep-plan office block – a 700 sq m gallery surrounded by 5,700 sq m of offices. There may be something in juxtaposing exhibition and work, but the 360-degree panorama of people sitting at desks makes for a dispiriting gallery-going experience.

Photograph: Hans Werlemann/courtesy OMA

“We wanted to delete the privacy between the tenants completely,” says Van Loon. “In a mixed-use building everyone wants to retreat to their own areas, but it’s important to stimulate new relationships and connections between them. There are many connecting doors here that are still locked.”

The original concept was more anarchic. “We had this idea of ‘architectural contamination’,” she says. “The DAC would steal space from the housing and offices, where they could demonstrate the future of housing or the workplace. We wanted the public pavement to go straight through the gym. But it was all deemed too radical.”

Where apartments were once scattered throughout the building, they now perch on the roof, arranged as a prosaic grid of 22 units that look on to a grim internal courtyard, all detailed with the same corporate hand, clad with dark metal grilles and green-tinged glass. They will be rented for around 25,000–30,000 kr (£2,900–£3,500) per month. For an institution that is opening with a thoughtful exhibition on new approaches to housing, it is ironic that it should have such poorly designed apartments on its own rooftop.

Photograph: Hans Werlemann/courtesy OMA

There are some powerful moments in the building, and fans of large cantilevers will be in awe. Arup engineer Chris Carroll (who also masterminded OMA’s daring CCTV building in Beijing), says the structure is even more complex than its Chinese cousin, comprising a stack of 18 vast steel trusses to create the expansive column-free spaces within. The raw steel sheet piling has also been left exposed in the lower levels, giving a forceful sense of holding back the water and contrasting with the shimmering brass floor in the Golden Gallery, a secondary space used for temporary exhibitions and precious loans. Outside, a corner of the building has been given over to a stepped playground, which does its best to be playful in the same cold palette of grey asphalt, steel and more mesh grillwork. It is OMA’s first ever playground, and it doesn’t look as if having fun comes naturally.

Hopefully, the combination of DAC and Bloxhub will be a boon for the industry, and a useful platform to get urban issues on the political agenda, but it feels as if the needs of the offices have trumped the ability to create a welcoming public building. It is a missed opportunity for Denmark, and all the more galling given that the country’s most sought-after architectural export, ex-OMA employee Bjarke Ingels, recently built a similarly sized stack of big staggered blocks in Billund for the Lego House – a joyous ziggurat that throws Copenhagen’s gloomy glass monolith into sharp relief.